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Math 55 chapter 11

Transcript of 11 Cylindrical and Spherical Coordinates - Handout

• Triple Integrals in Cylindrical& Spherical Coordinates

Math 55 - Elementary Analysis III

Institute of MathematicsUniversity of the Philippines

Diliman

Math 55 Cylindrical & Spherical Coordinates 1/ 23

• Cylindrical Coordinates

Consider a point P in space. Let Q be the projection of P ontothe xy-plane. Let r be the distance of Q from the origin, and be the angle between the positive x-axis and the line segmentOQ.

Qr

x

y

z

P

O

The ordered triple (r, , z) is called the cylindrical coordinatesof P .

Math 55 Cylindrical & Spherical Coordinates 2/ 23

• Surfaces in Cylindrical Coordinates

If r = c, a constant then we have a cylinder with radius |c|.

Math 55 Cylindrical & Spherical Coordinates 3/ 23

• Surfaces in Cylindrical Coordinates

If = c, a constant then we have a half-plane.

Math 55 Cylindrical & Spherical Coordinates 4/ 23

• Surfaces in Cylindrical Coordinates

If z = r, then we have a cone.

Math 55 Cylindrical & Spherical Coordinates 5/ 23

• Cartesian Cylindrical

r

x

y

z

P

O

The relationship between the Cartesian coordinates P (x, y, z)and cylindrical coordinates (r, , z) is given by

x = r cos

y = r sin

z = zMath 55 Cylindrical & Spherical Coordinates 6/ 23

• Cartesian Cylindrical

Remark

1 By the distance formula, r2 = x2 + y2.

2 dV = dz dA = rdz dr d

Math 55 Cylindrical & Spherical Coordinates 7/ 23

• Exercises

1 Write the following equations in cylindrical coordinates.

a. z =

3x2 +

3y2

b. x2 + y2 = 2y

c. z2 = 1 + x2 + y2

d. 2x2 + 2y2 + z2 = 4

2 A solid E lies within the cylinder x2 + y2 = 1, below theplane z = 4, and above the paraboloid z = 1 x2 y2.Find the volume of the solid E.

3 Evaluate

22

4y24y2

2x2+y2

xz dz dx dy

Math 55 Cylindrical & Spherical Coordinates 8/ 23

• Spherical Coordinates

Consider a point P in space. Let be the distance of P fromthe origin, be the same angle as in cylindrical coordinates, and be the angle between the positive z-axis and the line segment

OP .

x

y

z

P

O

The ordered triple (, , ) is called the spherical coordinates ofP .

Math 55 Cylindrical & Spherical Coordinates 9/ 23

• Surfaces in Spherical Coordinates

If = c, a constant then we have a sphere.

Math 55 Cylindrical & Spherical Coordinates 10/ 23

• Surfaces in Spherical Coordinates

If = c, a constant then we have a half-plane.

Math 55 Cylindrical & Spherical Coordinates 11/ 23

• Surfaces in Spherical Coordinates

If = c, a constant then we have a cone.

Figure: 0 c pi2 Figure: pi2 c pi

Math 55 Cylindrical & Spherical Coordinates 12/ 23

• Spherical CartesianSuppose P (x, y, z) has Spherical coordinates (, , ).

z

rx

y

P

Then z = cos. Let r be as in cylindrical coordinates. Thenr = sin. Therefore, x = r cos = sin cos andy = r sin = sin sin

Math 55 Cylindrical & Spherical Coordinates 13/ 23

• Spherical Cartesian

The relationship between the Cartesian coordinates P (x, y, z)and spherical coordinates (, , ) is given by

x = sin cos

y = sin sin

z = cos.

Also, the distance formula shows

2 = x2 + y2 + z2.

Note that > 0 and 0 pi.

Math 55 Cylindrical & Spherical Coordinates 14/ 23

• Spherical CartesianExample

Find the Cartesian coordinates of the point with sphericalcoordinates

(2, pi4 ,

pi3

).

Solution.

x = sin cos = 2(

sinpi

3

) (cos

pi

4

)= 2

(3

2

) (2

2

)=

6

2

y = sin sin = 2(

sinpi

3

)(sin

pi

4

)= 2

(3

2

)(2

2

)=

6

2

z = cos = 2(

cospi

3

)= 2

(1

2

)= 1.

Math 55 Cylindrical & Spherical Coordinates 15/ 23

• Spherical Cartesian

The point with spherical coordinates(2, pi4 ,

pi3

)is the point with

Cartesian coordinates (62 ,62 , 1).

Math 55 Cylindrical & Spherical Coordinates 16/ 23

• Spherical CartesianExample

A point P has Cartesian coordinates (0, 2

3,2). Find thespherical coordinates of P .

Solution. =

x2 + y2 + z2

=

0 + 12 + 4 =

16 = 4

cos =z

=24

= 12

=2pi

3,4pi

3

2pi

3

cos =x

sin= 0

=pi

2,3pi

2

pi

2because 0 pi since y > 0 Therefore, the sphericalcoordinates of P are (4, pi2 ,

2pi3 ).

Math 55 Cylindrical & Spherical Coordinates 17/ 23

• Triple Integrals in Spherical Coordinates

Suppose E is a solid given in spherical coordinates by

E = {(, , )|a b, , c d}

where a 0, 2pi and d c pi. We call this aspherical wedge.

We subdivide E into smaller sub-wedges Eijk by means ofequally spaced spheres = i, half-planes = j and half-cones = k.

Math 55 Cylindrical & Spherical Coordinates 18/ 23

• Triple Integrals in Spherical Coordinates

Consider the sub-wedge Eijk.

The volume of thesub-wedge can beapproximated by arectangular box withdimensions

, i, i sink

If Vijk is the volume ofEijk, then

Vijk 2i sink

Math 55 Cylindrical & Spherical Coordinates 19/ 23

• Triple Integrals in Spherical Coordinates

Suppose (i, j , k) has Cartesian coordinates (xijk, y

ijk, z

ijk).

ThenE

f(x, y, z)dV = liml,m,n

li=1

mj=1

nk=1

f(xijk, yijk, z

ijk)Vijk

which gives the formulaE

f(x, y, z)dV =

dc

ba

f( sin cos , sin sin , cos)2 sinddd

where E is the spherical wedge given by

E = {(, , )|a b, , c d}

Math 55 Cylindrical & Spherical Coordinates 20/ 23

• Triple Integrals in Spherical Coordinates

Example

Evaluate

E

z dV where E lies between the spheres

x2 + y2 + z2 = 1 and x2 + y2 + z2 = 4 in the first octant.

Solution. The equations of the spheres in sphericalcoordinates are = 1 and = 2. Since E is in the first octant,

E ={

(, , ) : 1 2, 0 pi2, 0 pi

2

}.

Hence,E

z dV =

pi2

0

pi2

0

21

( cos)2 sinddd

=

pi2

0

pi2

0

213 cos sinddd =

15pi

16

Math 55 Cylindrical & Spherical Coordinates 21/ 23

• Exercises

1 Write the following equations in spherical coordinates.

a. x = yb. z2 = x2 + y2

c. x2 + z2 = 9d. x2 + y2 + z2 = 2z

2 Evaluate

B

e(x2+y2+z2)

32 dV where B is the unit ball

centered at the origin.

3 Use spherical coordinates to find the volume of the solidthat lies above the cone z =

x2 + y2 and below the

sphere x2 + y2 + z2 = z.

4 Evaluate

30

9y20

18x2y2x2+y2

(x2 + y2 + z2) dz dx dy.

5 Find the mass of a solid E with constant density if E liesabove the cone z =

x2 + y2 and below the sphere

x2 + y2 + z2 = 1.

Math 55 Cylindrical & Spherical Coordinates 22/ 23

• References

1 Stewart, J., Calculus, Early Transcendentals, 6 ed., ThomsonBrooks/Cole, 2008

2 Dawkins, P., Calculus 3, online notes available atwww.lamar.com/pauldawkins

Math 55 Cylindrical & Spherical Coordinates 23/ 23